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Mirrors > Home > MPE Home > Th. List > mulgfvi | Structured version Visualization version Unicode version |
Description: The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
mulgfvi.t | .g |
Ref | Expression |
---|---|
mulgfvi | .g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgfvi.t | . 2 .g | |
2 | fvi 6255 | . . . . 5 | |
3 | 2 | eqcomd 2628 | . . . 4 |
4 | 3 | fveq2d 6195 | . . 3 .g .g |
5 | fvprc 6185 | . . . 4 .g | |
6 | fvprc 6185 | . . . . . 6 | |
7 | 6 | fveq2d 6195 | . . . . 5 .g .g |
8 | base0 15912 | . . . . . . . 8 | |
9 | eqid 2622 | . . . . . . . 8 .g .g | |
10 | 8, 9 | mulgfn 17544 | . . . . . . 7 .g |
11 | xp0 5552 | . . . . . . . 8 | |
12 | 11 | fneq2i 5986 | . . . . . . 7 .g .g |
13 | 10, 12 | mpbi 220 | . . . . . 6 .g |
14 | fn0 6011 | . . . . . 6 .g .g | |
15 | 13, 14 | mpbi 220 | . . . . 5 .g |
16 | 7, 15 | syl6eq 2672 | . . . 4 .g |
17 | 5, 16 | eqtr4d 2659 | . . 3 .g .g |
18 | 4, 17 | pm2.61i 176 | . 2 .g .g |
19 | 1, 18 | eqtri 2644 | 1 .g |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1483 wcel 1990 cvv 3200 c0 3915 cid 5023 cxp 5112 wfn 5883 cfv 5888 cz 11377 .gcmg 17540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-neg 10269 df-z 11378 df-seq 12802 df-slot 15861 df-base 15863 df-mulg 17541 |
This theorem is referenced by: (None) |
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